Mathematical Correfpondence. lg- 



manner, the probability of his fuccceding the third time is — , the fourth time ^, the fifth 

 8 , 6 49 +« 



-, the fixth 1, the feventh -, the eighth i-, the ninth 1, the tenth -i, the eleventh 

 ^' " *+:> 't^ 43 4- 



-, and the twelfth -1. And fince the probability of the happening of any number of events, 

 is equal to the produdl of the feveral probabilities of each of them happening, when con- 

 fidered feparately, we fliall have -x— v — v^v^vlv^v?. + 

 '-' SO "^ 49 "^ 48 ^ 47 ^ 46 ^ 4i ^ ff ^ « 

 ^ ^2 ^ ^ ^ ^ == i587533[i"9^ ^""^ '''^ probability required. So that the chances 

 againft the dealer, at the game of whift, having all the 13 trumps in his own hand, are above 

 a hundred thoufand million to one. And if the chances thus determined be fubftituted 

 in the analytical formula, given by Demoivre, in his Doarine of Chances, 3d edit. Prob. Ill, 

 it will be eafy to {hew that the number of trials in which this event may probably take place' 

 is 11112737229 ; whence, reckoning every game, one with another, to take 10 minutes, it 

 would be necefTary, in order to make it an equal chance for the event happening or failing, 

 that the gamefters fhbuld play on, uninterruptedly, for above two hundred thoufand years. ' 



Question IV, Anfwtred by Analyticvs. 



IT is laid down, by Newton, as a law of refrigeration, that the quantities of heat ba, in 

 given fmall fpaces of time, are always proportional to the heat remaining in the body ; or that 

 if the times be taken in arithmetical progreffion, the remaining heats will be in geometrical 

 progreffion ; reckoning thcfe heats to be the excefs by which the body is warmer than the 

 furrounding atmofphere. 



Suppofe, therefore, that h is the temperature of any body above that of the atmofphere 

 and that it cools d degrees in one minute. Then, by the above law, its temperature at the' 

 end of one, two, three, &c. minutes, will ht h — d, h^^ {^^), h^^d C~) &c. or 

 h — d,h {L—) , h I^JZ-) &c. Whence, unlverfally, Its temperature at the end of /« 

 minutes will be h {—}'", and at the end of « minutes h f^-'^)" 



And if thefe temperatures be determined by experiment, and denoted by a and b re- 

 fpeaively, we ftall have /. ('i^)" =. a, and /. (^-T-^" = b ., f^om which equations h is 



found = f_ 



, or log. h ^ —i-- {n log. a — m log. b), and rf = /. X (i — ij" '") 



^ «" " ~ h"-'" .., h-d *)"-'» 



^ 7 ■'^"" --f- = J = r = rate of cooling. 



Bb J 



Again, 



